The Conditions for Biot-Savart Law to Ampere's Law

[tade]

Mathematically, what conditions must a B-field that obeys the Biot-Savart Law satisfy before it will obey Ampere's Law?

Additionally, what conditions must the B-field obey in order to satisfy Faraday's Law?

 

[vanhees71]

The Biot-Savart and Ampere's Law are equivalent. The condition for their validity is that (a) all fields are time-independent and (b) that the stationary current obeys the (reduced) continuity equation ##\vec{\nabla} \cdot \vec{j}=0##. See

http://th.physik.uni-frankfurt.de/~hees/publ/ampere-law-discussion-ver2.pdf
http://dx.doi.org/10.1088/0143-0807/35/5/058001

 

[tade]

The Biot-Savart and Ampere's Law are equivalent. The condition for their validity is that (a) all fields are time-independent and (b) that the stationary current obeys the (reduced) continuity equation ##\vec{\nabla} \cdot \vec{j}=0##. See

http://th.physik.uni-frankfurt.de/~hees/publ/ampere-law-discussion-ver2.pdf
http://dx.doi.org/10.1088/0143-0807/35/5/058001

thanks. I wanted to apply this to Faraday's Law since it also follows Stokes' theorem.

 

[vanhees71]

Of course, you can do that, but that leads to very cumbersome and not very physically intuitive non-local equations. The local representation is given by Jefimenko's equation rather than by lumping field parts into the sources. It's also clear from relativistic covariance, that the electromagnetic field builds one "unit" in the sense that the electric and magnetic components together are the components of the electromagnetic field-strength (or Faraday) tensor ##F_{\mu \nu}## and the charge density and current density together are components of a four-vector field ##j^{\mu}=(c \rho,\vec{j})##.

 

[physiclawsrule]

Both laws have applications in aerodynamics for calculating the velocity induced by vortex lines by using the magnetic induction current formula B=µH